# Expectation of multivariate log normal

I am trying to compute the following:

Assume that $(a, b, c)$ are jointly normal with unknown means $(\mu_a, \mu_b, \mu_c)$ and variance-covariance matrix $\Sigma$.

What is $E[exp(a)|a>0, b>0, c>0]$?

When I input this into Mathematica, it hangs and fails to produce a result. My sense is that this is because I'm not telling it that $\Sigma$ is positive-definite (obviously it can tell that $\Sigma$ is symmetric).

My code is

Expectation[Exp[a] \[Conditioned] {a > 0, b > 0, c > 0},
{a, b, c} \[Distributed]
MultinormalDistribution[{μa, μb, μc},
{{σaa, σab, σac}, {σab, σbb, σbc}, {σac, σbc, σcc}}]]


Following the suggestion of using TruncatedDistribution, I tried running the following:

Expectation[Exp[x],
x \[Distributed]
MarginalDistribution[
TruncatedDistribution[{{0, ∞}, {0, ∞}, {0, ∞}},
MultinormalDistribution[{μa, μb, μc},
{{s11, s12, s13}, {s12, s22, s23}, {s13, s23, s33}}]], 1]]

• Please include your Mathematica code. – JimB Mar 4 '17 at 0:50
• What code did you use for the two-variable case? – David G. Stork Mar 4 '17 at 0:50
• Yep. That's why I deleted my comment. – David G. Stork Mar 4 '17 at 0:55
• Updated to include code – Shffl Mar 4 '17 at 0:57

Since you're only interested in the $x$ variable, form the Marginal Distribution by (effectively) integrating over $y$ and $z$. Then either do an integration over positive $x$:

Integrate[
x PDF[MarginalDistribution[
MultinormalDistribution[{μa, μb, μc},
{{s11, s12, s13}, {s12, s22, s23}, {s13, s23, s33}}], 1], x],
{x, 0, ∞}]

(*
ConditionalExpression[(
E^(-(μa^2/(2 s11))) s11 +
Sqrt[π/2] Sqrt[
s11] μa (1 + Erf[μa/(Sqrt Sqrt[s11])]))/(
Sqrt[2 π] Sqrt[s11]), (Re[s11] >= 0 && Re[μa/s11] < 0) ||
Re[s11] > 0]
*)


or a one-dimensional Expectation over the new distribution.

Basic sanity check:

Expectation[x,
x \[Distributed]
MarginalDistribution[
MultinormalDistribution[{μa, μb, μc}, {{s11, s12,
s13}, {s12, s22, s23}, {s13, s23, s33}}], 1]]

(*
μa
*)


So here is nearly the final answer (which does not constrain $y$ and $z$ to be positive):

Expectation[Exp[x] \[Conditioned] x > 0,
x \[Distributed]
MarginalDistribution[
MultinormalDistribution[{μa, μb, μc},
{{s11, s12, s13}, {s12, s22, s23}, {s13, s23, s33}}], 1]]


$\frac{e^{\text{$\mu $a}+\frac{\text{s11}}{2}} \left(\text{erf}\left(\frac{\text{$\mu $a}+\text{s11}}{\sqrt{2} \sqrt{\text{s11}}}\right)+1\right)}{\text{erfc}\left(-\frac{\text{$\mu $a}}{\sqrt{2} \sqrt{\text{s11}}}\right)}$

I think to constrain $y$ and $z$ to be positive you'll have to use TruncatedDistribution. (I may have time to solve this tomorrow...)

• While people have voted for this to be the answer, it doesn't handle the constrained case, which is the important one, as the marginal distribution of a multivariate normal is easy to get. – Shffl Mar 5 '17 at 20:06